
respectively, provided the right-hand side is well defined as a finite number or \(\pm \infty \)  (for more details see [1]}). For \(\mu =\nu \) , \(\kappa (\mu ,\nu )\)  defines the energy \(\kappa (\mu ,\mu )\)  of \(\mu \) .
In particular, if the measures are positive, then \(\kappa \mu (x)\) , resp. \(\kappa (\mu ,\nu )\) , is always well defined and represents a l.s.c. function of \((x,\mu )\in X\times \mathfrak {M}^+\) , resp. of \((\mu ,\nu )\in \mathfrak {M}^+\times \mathfrak {M}^+\)  (the principle of descent [1]}, cf. Lemma REF  and footnote REF ).
